IIT-JEE · Practice Sheet 15

1. 1.

   Mathematics

   If f(x) = 3x³ + 3x² + 8x, find f′(3).

   Formula:Derivative of a Polynomial

   $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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2. 2.

   Mathematics

   Find the distance between the points (-5, -3) and (2, -8).

   Formula:Distance Between Two Points

   $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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3. 3.

   Quantitative Aptitude

   In how many ways can 4 objects be chosen from 5 distinct objects? (Find ⁿᴄᵣ for n = 5, r = 4.)

   Formula:Combinations

   $^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}$

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4. 4.

   Physics

   Find the momentum of a 20 kg body moving at 36 m/s.

   Formula:Linear Momentum

   $p = mv$

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5. 5.

   Physics

   Find the force required to accelerate a 22 kg mass at 14 m/s².

   Formula:Newton's Second Law (F = ma)

   $F = ma$

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6. 6.

   Mathematics

   Evaluate log base 2 of 32.

   Formula:Logarithm Evaluation

   $\log_{b}(b^{k}) = k$

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7. 7.

   Physics

   A projectile is launched at 45 m/s at 45° to the horizontal. Find its range (g = 9.8 m/s²).

   Formula:Projectile Range

   $R = \dfrac{u^{2}\sin 2\theta}{g}$

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8. 8.

   Chemistry

   A solution is made by dissolving 28 g of solute in 199 g of solvent. Find the mass percentage of the solute.

   Formula:Mass Percentage

   $\%\,\text{mass} = \dfrac{\text{solute}}{\text{solution}}\times 100$

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9. 9.

   Mathematics

   If f(x) = 4x³ + 1x² + 5x, find f′(5).

   Formula:Derivative of a Polynomial

   $\dfrac{d}{dx}x^{n} = n\,x^{\,n-1}$

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10. 10.

    Mathematics

    Find term number 7 of a geometric progression with first term 6 and common ratio 3.

    Formula:GP nth Term

    $a_n = a\,r^{\,n-1}$

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11. 11.

    Physics

    Find the force required to accelerate a 36 kg mass at 17 m/s².

    Formula:Newton's Second Law (F = ma)

    $F = ma$

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12. 12.

    Mathematics

    Find the slope of the line through (-8, -2) and (-2, 6).

    Formula:Slope of a Line

    $m = \dfrac{y_2-y_1}{x_2-x_1}$

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13. 13.

    Chemistry

    A solution is made by dissolving 23 g of solute in 94 g of solvent. Find the mass percentage of the solute.

    Formula:Mass Percentage

    $\%\,\text{mass} = \dfrac{\text{solute}}{\text{solution}}\times 100$

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14. 14.

    Mathematics

    Find the distance between the points (7, -1) and (-2, -3).

    Formula:Distance Between Two Points

    $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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15. 15.

    Quantitative Aptitude

    In how many ways can 3 objects be chosen from 5 distinct objects? (Find ⁿᴄᵣ for n = 5, r = 3.)

    Formula:Combinations

    $^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}$

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16. 16.

    Chemistry

    Find the pressure of 5 mol of an ideal gas at 211 K in a 10 L container (R = 0.0821 L·atm·mol⁻¹·K⁻¹).

    Formula:Ideal Gas Law

    $PV = nRT$

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17. 17.

    Mathematics

    For the quadratic 4x² + 4x + 2 = 0, find the sum and product of its roots.

    Formula:Vieta's Formulas (roots)

    $\alpha+\beta=-\dfrac{b}{a},\;\alpha\beta=\dfrac{c}{a}$

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18. 18.

    Chemistry

    Find the molarity of a solution containing 0.5 mol of solute in 2 L of solution.

    Formula:Molarity

    $M = \dfrac{n}{V}$

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19. 19.

    Mathematics

    Find the binomial coefficient C(10, 4) — the coefficient of x^4 in (1 + x)^10.

    Formula:Binomial Coefficient

    $\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}$

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20. 20.

    Chemistry

    How many moles are present in 110 g of carbon dioxide (CO₂) (molar mass = 44 g/mol)?

    Formula:Number of Moles

    $n = \dfrac{m}{M}$

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